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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Gaussian rendering</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">This method draws a normalized symmetric 2D Gaussian function integrated
over the voxel volume for every localized molecule, with a standard
deviation equal to the computed, or user-specified localization uncertainty.
The visualized molecules are added sequentially to the final super-resolution
images. The contribution of one molecule to the voxel intensity at
the integer position <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi26.png" width="62" height="21" alt="\left(x,y,z\right)"> is computed as
</p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi18.png" width="203" height="22" alt="v\left(x,y,z\mid\boldsymbol{\theta}_{p}\right)=E_{x}E_{y}E_{z}\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
<p class="ltx_p">where <img id="Sx1.p1.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi28.png" width="13" height="15" alt="p"> indexes the molecules, and the parameters <img id="Sx1.p1.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi22.png" width="206" height="22" alt="\boldsymbol{\theta}_{p}=\left[\hat{x}_{p},\hat{y}_{p},\hat{z}_{p},\hat{\sigma}%
_{xy,p},\hat{\sigma}_{z,p}\right]">.
Here <img id="Sx1.p1.m4" class="ltx_Math" style="vertical-align:-7px" src="mi/mi25.png" width="71" height="21" alt="\hat{x}_{p},\hat{y}_{p},\hat{z}_{p}"> is the estimated position
of a molecule, <img id="Sx1.p1.m5" class="ltx_Math" style="vertical-align:-7px" src="mi/mi23.png" width="42" height="21" alt="\hat{\sigma}_{xy,p}"> is the corresponding lateral
localization uncertainty, <img id="Sx1.p1.m6" class="ltx_Math" style="vertical-align:-7px" src="mi/mi24.png" width="34" height="21" alt="\hat{\sigma}_{z,p}"> is the axial localization
uncertainty,</p>
<table id="Sx1.EGx1" class="ltx_equationgroup ltx_eqn_eqnarray">

<tr id="Sx1.E2" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi10.png" width="26" height="18" alt="\displaystyle E_{x}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.E2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi8.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.E2.m3" class="ltx_Math" style="vertical-align:-24px" src="mi/mi15.png" width="330" height="57" alt="\displaystyle\frac{1}{2}\erf\left(\frac{x-\hat{x}+\frac{1}{2}}{\sqrt{2}\hat{%
\sigma}_{xy}}\right)-\frac{1}{2}\erf\left(\frac{x-\hat{x}-\frac{1}{2}}{\sqrt{2%
}\hat{\sigma}_{xy}}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
<tr id="Sx1.E3" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.E3.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi12.png" width="26" height="21" alt="\displaystyle E_{y}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.E3.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi8.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.E3.m3" class="ltx_Math" style="vertical-align:-24px" src="mi/mi16.png" width="326" height="57" alt="\displaystyle\frac{1}{2}\erf\left(\frac{y-\hat{y}+\frac{1}{2}}{\sqrt{2}\hat{%
\sigma}_{xy}}\right)-\frac{1}{2}\erf\left(\frac{y-\hat{y}-\frac{1}{2}}{\sqrt{2%
}\hat{\sigma}_{xy}}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(3)</span></td>
</tr>
<tr id="Sx1.E4" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.E4.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi14.png" width="25" height="18" alt="\displaystyle E_{z}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.E4.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi8.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.E4.m3" class="ltx_Math" style="vertical-align:-24px" src="mi/mi17.png" width="349" height="57" alt="\displaystyle\frac{1}{2}\erf\left(\frac{z-\hat{z}+\frac{\Delta_{z}}{2}}{\sqrt{%
2}\hat{\sigma}_{z}}\right)-\frac{1}{2}\erf\left(\frac{z-\hat{z}-\frac{\Delta_{%
z}}{2}}{\sqrt{2}\hat{\sigma}_{z}}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(4)</span></td>
</tr>
</table>
<p class="ltx_p">and <img id="Sx1.p1.m7" class="ltx_Math" style="vertical-align:-5px" src="mi/mi27.png" width="28" height="19" alt="\triangle_{z}"> is the size of a voxel in the axial direction.
Contributions from one molecule are limited to an interval given by
a circle with radius of <img id="Sx1.p1.m8" class="ltx_Math" style="vertical-align:-7px" src="mi/mi19.png" width="51" height="21" alt="3\hat{\sigma}_{xy,p}"> around the molecule
position in lateral dimension and by <img id="Sx1.p1.m9" class="ltx_Math" style="vertical-align:-7px" src="mi/mi20.png" width="43" height="21" alt="3\hat{\sigma}_{z,p}"> in axial
direction. For data visualization in the 2D case, <img id="Sx1.p1.m10" class="ltx_Math" style="vertical-align:-2px" src="mi/mi29.png" width="45" height="16" alt="z=0"> and the term
<img id="Sx1.p1.m11" class="ltx_Math" style="vertical-align:-5px" src="mi/mi21.png" width="57" height="18" alt="E_{z}=1">.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Rendering.html" title="" class="ltx_ref">Visualization methods</a></p>
</div>
</li>
</ul>
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